{"id":449,"date":"2019-01-27T20:18:24","date_gmt":"2019-01-27T11:18:24","guid":{"rendered":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/linear-regression-analysis-using-matrix-in-r\/"},"modified":"2024-10-12T10:56:46","modified_gmt":"2024-10-12T01:56:46","slug":"linear-regression-analysis-using-matrix-in-r","status":"publish","type":"post","link":"https:\/\/best-biostatistics.com\/toukei-er\/entry\/linear-regression-analysis-using-matrix-in-r\/","title":{"rendered":"R \u3067\u884c\u5217\u8a08\u7b97\u3092\u4f7f\u3063\u3066\u91cd\u56de\u5e30\u5206\u6790\u3092\u884c\u3046\u65b9\u6cd5"},"content":{"rendered":"\n

\u91cd\u56de\u5e30\u5206\u6790\u306f\u884c\u5217\u8a08\u7b97\u3092\u3057\u3066\u3044\u308b\u308f\u3051\u3060\u304c\u3001\u5b9f\u969b\u3069\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u3057\u3066\u3044\u308b\u306e\u304b\u3001R \u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n\n\n\n\n

\u30b5\u30f3\u30d7\u30eb\u30c7\u30fc\u30bf\u306e\u6e96\u5099<\/h2>\n\n\n\n

\u4f7f\u7528\u3059\u308b\u30c7\u30fc\u30bf\u306f ISLR \u30d1\u30c3\u30b1\u30fc\u30b8\u306e Carseats \u30c7\u30fc\u30bf\u3002<\/p>\n\n\n\n

\u3053\u308c\u306f\u30c1\u30e3\u30a4\u30eb\u30c9\u30b7\u30fc\u30c8\u306e\u58f2\u308a\u4e0a\u3052\u30c7\u30fc\u30bf\u3002<\/p>\n\n\n\n

\u30d1\u30c3\u30b1\u30fc\u30b8\u306e\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u65b9\u6cd5\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n

install.packages<\/span>(<\/span>'ISLR'<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u30d1\u30c3\u30b1\u30fc\u30b8\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u306e\u904e\u53bb\u8a18\u4e8b\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n
\t\t\t
\n\t\t\t\t
\n\t\t\t\t\t\u3042\u308f\u305b\u3066\u8aad\u307f\u305f\u3044<\/span>\n\t\t\t\t\t
\"\"<\/figure><\/div>\t\t\t\t\t
\n\t\t\t\t\t\tR \u306b\u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3059\u308b\u65b9\u6cd5<\/a>\n\t\t\t\t\t\tR\u306f\u3001\u6700\u521d\u304b\u3089\u304b\u306a\u308a\u3044\u308d\u3044\u308d\u306a\u3053\u3068\u304c\u3067\u304d\u308b\u7121\u6599\u7d71\u8a08\u30bd\u30d5\u30c8\u3002 \u3082\u3063\u3068\u3059\u3054\u3044\u306e\u306f\u3001\u3042\u3068\u304b\u3089\u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3057\u3066\u3001\u3055\u3089\u306b\u3044\u308d\u3044\u308d\u306a\u89e3\u6790\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3053\u3068…<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\n\n
\u30c7\u30fc\u30bf\u30bb\u30c3\u30c8\u5185\u306e\u9023\u7d9a\u30c7\u30fc\u30bf\u306e\u307f\u3092\u6271\u3046\u3002<\/p>\n\n\n\n
library<\/span>(<\/span>ISLR)<\/span>\nstr<\/span>(<\/span>Carseats)<\/span>\nCarseats2 <-<\/span> Carseats[,<\/span>c<\/span>(<\/span>1<\/span>:<\/span>6<\/span>,<\/span>8<\/span>,<\/span>9<\/span>)]<\/span>\nstr<\/span>(<\/span>Carseats)<\/span>\n<\/code><\/pre>\n\n\n\n
\u4f7f\u7528\u3059\u308b\u30c7\u30fc\u30bf\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
Sales\u3092\u307b\u304b\u306e7\u3064\u306e\u5909\u6570\u3067\u4e88\u6e2c\u3059\u308b\u91cd\u56de\u5e30\u30e2\u30c7\u30eb\u3092\u4f5c\u308b\u3002<\/p>\n\n\n\n
><\/span> str<\/span>(<\/span>Carseats2)<\/span>\n'data.frame'<\/span>:<\/span>   400<\/span> obs. of  8<\/span> variables:<\/span>\n$<\/span> Sales      :<\/span> num  9.5<\/span> 11.22<\/span> 10.06<\/span> 7.4<\/span> 4.15<\/span> ...\n$<\/span> CompPrice  :<\/span> num  138<\/span> 111<\/span> 113<\/span> 117<\/span> 141<\/span> 124<\/span> 115<\/span> 136<\/span> 132<\/span> 132<\/span> ...\n$<\/span> Income     :<\/span> num  73<\/span> 48<\/span> 35<\/span> 100<\/span> 64<\/span> 113<\/span> 105<\/span> 81<\/span> 110<\/span> 113<\/span> ...\n$<\/span> Advertising:<\/span> num  11<\/span> 16<\/span> 10<\/span> 4<\/span> 3<\/span> 13<\/span> 0<\/span> 15<\/span> 0<\/span> 0<\/span> ...\n$<\/span> Population :<\/span> num  276<\/span> 260<\/span> 269<\/span> 466<\/span> 340<\/span> 501<\/span> 45<\/span> 425<\/span> 108<\/span> 131<\/span> ...\n$<\/span> Price      :<\/span> num  120<\/span> 83<\/span> 80<\/span> 97<\/span> 128<\/span> 72<\/span> 108<\/span> 120<\/span> 124<\/span> 124<\/span> ...\n$<\/span> Age        :<\/span> num  42<\/span> 65<\/span> 59<\/span> 55<\/span> 38<\/span> 78<\/span> 71<\/span> 67<\/span> 76<\/span> 76<\/span> ...\n$<\/span> Education  :<\/span> num  17<\/span> 10<\/span> 12<\/span> 14<\/span> 13<\/span> 16<\/span> 15<\/span> 10<\/span> 10<\/span> 17<\/span> ...\n<\/code><\/pre>\n\n\n\n
lm() \u3067\u8a08\u7b97\u3057\u3066\u307f\u308b<\/h2>\n\n\n\n
\u884c\u5217\u3092\u4f7f\u3063\u3066 Step by Step \u3067\u8a08\u7b97\u3057\u3066\u307f\u308b\u524d\u306b\u3001lm() \u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n
lm2 <-<\/span> lm<\/span>(<\/span>Sales~<\/span>.,<\/span>data=<\/span>Carseats2)<\/span>\nsummary<\/span>(<\/span>lm2)<\/span>\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
><\/span> summary<\/span>(<\/span>lm2)<\/span>\nCall:<\/span>\nlm<\/span>(<\/span>formula =<\/span> Sales ~<\/span> .,<\/span> data =<\/span> Carseats2)<\/span>\nResiduals:<\/span>\nMin      1Q  Median      3Q     Max\n-<\/span>5.0598<\/span> -<\/span>1.3515<\/span> -<\/span>0.1739<\/span>  1.1331<\/span>  4.8304<\/span>\nCoefficients:<\/span>\nEstimate Std. Error t value Pr<\/span>(<\/span>>|<\/span>t|<\/span>)<\/span>\n(<\/span>Intercept)<\/span>  7.7076934<\/span>  1.1176260<\/span>   6.896<\/span> 2.15e-11<\/span> ***<\/span>\nCompPrice    0.0939149<\/span>  0.0078395<\/span>  11.980<\/span>  <<\/span> 2e-16<\/span> ***<\/span>\nIncome       0.0128717<\/span>  0.0034757<\/span>   3.703<\/span> 0.000243<\/span> ***<\/span>\nAdvertising  0.1308637<\/span>  0.0151219<\/span>   8.654<\/span>  <<\/span> 2e-16<\/span> ***<\/span>\nPopulation  -<\/span>0.0001239<\/span>  0.0006877<\/span>  -<\/span>0.180<\/span> 0.857092<\/span>\nPrice       -<\/span>0.0925226<\/span>  0.0050521<\/span> -<\/span>18.314<\/span>  <<\/span> 2e-16<\/span> ***<\/span>\nAge         -<\/span>0.0449743<\/span>  0.0060083<\/span>  -<\/span>7.485<\/span> 4.75e-13<\/span> ***<\/span>\nEducation   -<\/span>0.0399844<\/span>  0.0371257<\/span>  -<\/span>1.077<\/span> 0.282142<\/span>\n---<\/span>\nSignif. codes:<\/span>  0<\/span> \u2018***<\/span>\u2019 0.001<\/span> \u2018**<\/span>\u2019 0.01<\/span> \u2018*<\/span>\u2019 0.05<\/span> \u2018.\u2019 0.1<\/span> \u2018 \u2019 1<\/span>\nResidual standard error:<\/span> 1.929<\/span> on 392<\/span> degrees of freedom\nMultiple R-<\/span>squared:<\/span>  0.5417<\/span>,<\/span>    Adjusted R-<\/span>squared:<\/span>  0.5335<\/span>\nF<\/span>-<\/span>statistic:<\/span> 66.18<\/span> on 7<\/span> and 392<\/span> DF,<\/span>  p-<\/span>value:<\/span> <<\/span> 2.2e-16<\/span>\n<\/code><\/pre>\n\n\n\n
\uff1e\uff1e\u3082\u3046\u7d71\u8a08\u3067\u60a9\u3080\u306e\u306f\u7d42\u308f\u308a\u306b\u3057\u307e\u305b\u3093\u304b\uff1f\u00a0<\/a><\/strong><\/span><\/p>\r\n\"\"<\/a>\r\n
\u21911\u4e07\u4eba\u4ee5\u4e0a\u306e\u533b\u7642\u5f93\u4e8b\u8005\u304c\u8cfc\u8aad\u4e2d<\/span><\/strong><\/span><\/p><\/div>
\u884c\u5217\u8a08\u7b97\u3067\u5b9f\u884c\u3059\u308b\u65b9\u6cd5<\/h2>\n\n\n\n
\u30c7\u30fc\u30bf\u3092\u884c\u5217\u306b\u3059\u308b<\/h3>\n\n\n\n
n\u3068p\u306f\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\u3068\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u6570\u3002<\/p>\n\n\n\n
\u30c7\u30fc\u30bf\u3092\u884c\u5217\u306b\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n\n\n\n
X\u306e\u6700\u521d\u306e\u5217\u306b\u5207\u7247\u3068\u3057\u3066\u3059\u3079\u3066\u306e\u30c7\u30fc\u30bf\u884c\u306b1\u3092\u5165\u529b\u3057\u305f\u3002<\/p>\n\n\n\n
\u3061\u306a\u307f\u306b[,1]\u306f\u4e00\u5217\u76ee\u306e\u307f\u3092\u53d6\u308a\u51fa\u3059\u6307\u793a\u3001[,-1]\u306f\u4e00\u5217\u76ee\u4ee5\u5916\u3092\u53d6\u308a\u51fa\u3059\u6307\u793a\u3002<\/p>\n\n\n\n
as.matrix() \u306f\u884c\u5217\u3068\u3057\u3066\u6271\u3046\u6307\u793a\u3002<\/p>\n\n\n\n
y <-<\/span> as.matrix<\/span>(<\/span>Carseats2[,<\/span>1<\/span>])<\/span>\nn <-<\/span> length<\/span>(<\/span>y)<\/span>\np <-<\/span> length<\/span>(<\/span>Carseats2[,<\/span>-<\/span>1<\/span>])<\/span>\nX <-<\/span> as.matrix<\/span>(<\/span>cbind<\/span>(<\/span>\"Intercept\"<\/span>=<\/span>rep<\/span>(<\/span>1<\/span>,<\/span>n),<\/span>Carseats2[,<\/span>-<\/span>1<\/span>]))<\/span>\nstr<\/span>(<\/span>y)<\/span>\nstr<\/span>(<\/span>X)<\/span>\n<\/code><\/pre>\n\n\n\n
y\u3068X\u306fstr()\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u898b\u3048\u308b\u3002<\/p>\n\n\n\n
><\/span> str<\/span>(<\/span>y)<\/span>\nnum [<\/span>1<\/span>:<\/span>400<\/span>,<\/span> 1<\/span>]<\/span> 9.5<\/span> 11.22<\/span> 10.06<\/span> 7.4<\/span> 4.15<\/span> ...\n><\/span> str<\/span>(<\/span>X)<\/span>\nnum [<\/span>1<\/span>:<\/span>400<\/span>,<\/span> 1<\/span>:<\/span>8<\/span>]<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> 1<\/span> ...\n-<\/span> attr<\/span>(<\/span>*<\/span>,<\/span> \"dimnames\"<\/span>)<\/span>=<\/span>List of 2<\/span>\n..$<\/span> :<\/span> chr [<\/span>1<\/span>:<\/span>400<\/span>]<\/span> \"1\"<\/span> \"2\"<\/span> \"3\"<\/span> \"4\"<\/span> ...\n..$<\/span> :<\/span> chr [<\/span>1<\/span>:<\/span>8<\/span>]<\/span> \"Intercept\"<\/span> \"CompPrice\"<\/span> \"Income\"<\/span> \"Advertising\"<\/span> ...\n<\/code><\/pre>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u30d9\u30af\u30c8\u30eb\u3092\u6c42\u3081\u308b\uff08\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u884c\u5217\u3067\uff09<\/h3>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u306e\u63a8\u5b9a\u5024\u30d9\u30af\u30c8\u30eb $ \\boldsymbol{\\hat{\\beta}} $ \u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u5f0f\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
$$ \\boldsymbol{\\hat{\\beta}} = (\\boldsymbol{X}^t \\boldsymbol{X})^{-1} \\boldsymbol{X}^t \\boldsymbol{y} $$<\/p>\n\n\n\n\n\n\n\n
R \u3067\u8a08\u7b97\u3057\u3066\u307f\u308b\u3002<\/p>\n\n\n\n
\u307e\u305a\u3001$ \\boldsymbol{X}^t \\boldsymbol{X} $ <\/p>\n\n\n\n
\u3053\u308c\u306f\u8ee2\u7f6e t() \u3068\u3001\u884c\u5217\u306e\u639b\u3051\u7b97%*%\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/p>\n\n\n\n
t<\/span>(<\/span>X)<\/span> %*%<\/span> X\n<\/code><\/pre>\n\n\n\n
\u3059\u308b\u3068\u3001\u884c\u5217\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\u304c\u914d\u7f6e\u3055\u308c\u305f8\u00d78\u306e\u884c\u5217\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
><\/span> t<\/span>(<\/span>X)<\/span> %*%<\/span> X\n            Intercept CompPrice  Income Advertising Population    Price     Age Education\nIntercept         400<\/span>     49990<\/span>   27463<\/span>        2654<\/span>     105936<\/span>    46318<\/span>   21329<\/span>      5560<\/span>\nCompPrice       49990<\/span>   6341324<\/span> 3418378<\/span>      330699<\/span>   13153953<\/span>  5873316<\/span> 2655656<\/span>    695265<\/span>\nIncome          27463<\/span>   3418378<\/span> 2198045<\/span>      186598<\/span>    7260338<\/span>  3165088<\/span> 1463551<\/span>    380072<\/span>\nAdvertising      2654<\/span>    330699<\/span>  186598<\/span>       35256<\/span>     806772<\/span>   310118<\/span>  141322<\/span>     36657<\/span>\nPopulation     105936<\/span>  13153953<\/span> 7260338<\/span>      806772<\/span>   36722296<\/span> 12249952<\/span> 5608130<\/span>   1456118<\/span>\nPrice           46318<\/span>   5873316<\/span> 3165088<\/span>      310118<\/span>   12249952<\/span>  5587066<\/span> 2454154<\/span>    644111<\/span>\nAge             21329<\/span>   2655656<\/span> 1463551<\/span>      141322<\/span>    5608130<\/span>  2454154<\/span> 1242033<\/span>    296583<\/span>\nEducation        5560<\/span>    695265<\/span>  380072<\/span>       36657<\/span>    1456118<\/span>   644111<\/span>  296583<\/span>     80024<\/span>\n<\/code><\/pre>\n\n\n\n
\u6b21\u306b\u3001\u9006\u884c\u5217 $ (\\boldsymbol{X}^t \\boldsymbol{X})^{-1} $ \u306f\u3069\u3046\u3084\u3063\u3066\u66f8\u304f\u304b\u3002<\/p>\n\n\n\n
\u3053\u308c\u306fsolve() \u3092\u4f7f\u3063\u3066\u6587\u5b57\u901a\u308a\u300c\u89e3\u304f\u300d\u3002<\/p>\n\n\n\n
\u7d50\u679c\u306f8×8\u306e\u884c\u5217\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n\n\n\n
solve<\/span>(<\/span>t<\/span>(<\/span>X)<\/span>%*%<\/span>X)<\/span>\n<\/code><\/pre>\n\n\n\n
\u3053\u306e\u9006\u884c\u5217\u306b\u53f3\u304b\u3089 $ \\boldsymbol{X} $ \u306e\u8ee2\u7f6e\u884c\u5217\u3092\u304b\u3051\u3066\u3001\u3055\u3089\u306b $\\boldsymbol{y} $ \u3092\u639b\u3051\u308b\u3068\u56de\u5e30\u4fc2\u6570\u30d9\u30af\u30c8\u30eb\u304c\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
solve<\/span>(<\/span>t<\/span>(<\/span>X)<\/span>%*%<\/span>X)<\/span> %*%<\/span> t<\/span>(<\/span>X)<\/span> %*%<\/span> y\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u3053\u3061\u3089\u3002<\/p>\n\n\n\n
\u3053\u308c\u304c\u3044\u308f\u3086\u308b\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3067\u3001\u884c\u5217\u3092\u4f7f\u3046\u3068\u3053\u3093\u306a\u8a08\u7b97\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
><\/span> solve<\/span>(<\/span>t<\/span>(<\/span>X)<\/span>%*%<\/span>X)<\/span> %*%<\/span> t<\/span>(<\/span>X)<\/span> %*%<\/span> y\n[,<\/span>1<\/span>]<\/span>\nIntercept    7.7076934384<\/span>\nCompPrice    0.0939149066<\/span>\nIncome       0.0128717129<\/span>\nAdvertising  0.1308636707<\/span>\nPopulation  -<\/span>0.0001239252<\/span>\nPrice       -<\/span>0.0925226099<\/span>\nAge         -<\/span>0.0449743402<\/span>\nEducation   -<\/span>0.0399844437<\/span>\n<\/code><\/pre>\n\n\n\n
\u3053\u308c\u306f\u3001lm() \u3092\u4f7f\u3063\u305f\u6642\u306ecoefficients\u306b\u4e00\u81f4\u3059\u308b\u3002<\/p>\n\n\n\n
Estimate\u306e\u5217\u306e\u6570\u5024\u304c\u4e00\u81f4\u3057\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b\u3002<\/p>\n\n\n\n
><\/span> summary<\/span>(<\/span>lm2)$<\/span>coefficients\n               Estimate   Std. Error     t value     Pr<\/span>(<\/span>>|<\/span>t|<\/span>)<\/span>\n(<\/span>Intercept)<\/span>  7.7076934384<\/span> 1.1176259965<\/span>   6.8964873<\/span> 2.145154e-11<\/span>\nCompPrice    0.0939149066<\/span> 0.0078395225<\/span>  11.9796718<\/span> 2.153866e-28<\/span>\nIncome       0.0128717129<\/span> 0.0034756701<\/span>   3.7033759<\/span> 2.432641e-04<\/span>\nAdvertising  0.1308636707<\/span> 0.0151219066<\/span>   8.6539134<\/span> 1.302560e-16<\/span>\nPopulation  -<\/span>0.0001239252<\/span> 0.0006877272<\/span>  -<\/span>0.1801952<\/span> 8.570924e-01<\/span>\nPrice       -<\/span>0.0925226099<\/span> 0.0050520870<\/span> -<\/span>18.3137403<\/span> 1.409811e-54<\/span>\nAge         -<\/span>0.0449743402<\/span> 0.0060082977<\/span>  -<\/span>7.4853715<\/span> 4.751078e-13<\/span>\nEducation   -<\/span>0.0399844437<\/span> 0.0371257460<\/span>  -<\/span>1.0770004<\/span> 2.821424e-01<\/span>\n<\/code><\/pre>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u30d9\u30af\u30c8\u30eb\u306b beta \u3068\u3044\u3046\u540d\u524d\u3092\u4ed8\u3051\u3066\u304a\u304f\u3002<\/p>\n\n\n\n
beta <-<\/span> solve<\/span>(<\/span>t<\/span>(<\/span>X)<\/span>%*%<\/span>X)<\/span> %*%<\/span> t<\/span>(<\/span>X)<\/span> %*%<\/span> y\n<\/code><\/pre>\n\n\n\n
\u4e88\u6e2c\u5024y.hat\u3092\u6c42\u3081\u3066\u3001\u5b9f\u6e2c\u5024y\u3068\u6563\u5e03\u56f3\u3092\u63cf\u3044\u3066\u307f\u308b<\/h3>\n\n\n\n
\u5b9f\u6e2c\u5024\u30d9\u30af\u30c8\u30eb $ \\boldsymbol{X} $ \u3068\u56de\u5e30\u4fc2\u6570\u30d9\u30af\u30c8\u30eb $ \\boldsymbol{\\hat{\\beta}} $ \u3092\u4f7f\u3063\u3066\u4e88\u6e2c\u5024 $ \\hat{y} $ \u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n
\u884c\u5217\u8868\u73fe\u3092\u3059\u308c\u3070\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
$$ \\boldsymbol{\\hat{y}} = \\boldsymbol{X \\beta} $$<\/p>\n\n\n\n\n\n\n\n
R \u3067\u306f\u3001\u4ee5\u4e0b\u306e\u901a\u308a\u306e\u30b9\u30af\u30ea\u30d7\u30c8\u306b\u306a\u308b\u3002<\/p>\n\n\n\n
\u4e88\u6e2c\u5024\u3092y.hat\u3068\u3057\u305f\u3002<\/p>\n\n\n\n
y.hat <-<\/span> X %*%<\/span> beta\n<\/code><\/pre>\n\n\n\n
\u5b9f\u6e2c\u5024 y \u3068\u4e88\u6e2c\u5024 y.hat \u306e\u6563\u5e03\u56f3\u3092\u63cf\u3044\u3066\u307f\u305f\u3002<\/p>\n\n\n\n
\u5207\u7247 0\u3001\u50be\u304d 1 \u306e\u76f4\u7dda y = x \u3082\u63cf\u304d\u5165\u308c\u305f\u3002<\/p>\n\n\n\n
\u5b9f\u6e2c\u5024\u3068\u4e88\u6e2c\u5024\u304c\u4e00\u81f4\u3059\u308c\u3070\u76f4\u7dda\u72b6\u306b\u4e26\u3076\u306f\u305a\u3060\u3002<\/p>\n\n\n\n
\u5b9f\u969b\u306b\u306f\u5b8c\u5168\u306b\u306f\u4e88\u6e2c\u3067\u304d\u306a\u3044\u306e\u3067\u3001\u76f4\u7dda\u72b6\u306b\u306f\u4e26\u3070\u306a\u3044\u3002<\/p>\n\n\n\n
plot<\/span>(<\/span>y.hat,<\/span>y,<\/span> xlim=<\/span>c<\/span>(<\/span>0<\/span>,<\/span>15<\/span>),<\/span>ylim=<\/span>c<\/span>(<\/span>0<\/span>,<\/span>15<\/span>))<\/span>\nabline<\/span>(<\/span>a=<\/span>0<\/span>,<\/span>b=<\/span>1<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
<\/span><\/p>\n\n\n\n
\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n
\u6b21\u306b\u3001\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u306e\u6e96\u5099\u3068\u3057\u3066\u3001\u5206\u6563\u5171\u5206\u6563\u884c\u5217\uff08V.beta\uff09\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n
V.beta\u306e\u8a08\u7b97\u306e\u305f\u3081\u306b\u3001\u8aa4\u5dee\u5206\u6563 sigmaE2 \u3092\u5148\u306b\u8a08\u7b97\u3057\u3066\u304a\u304f\u3002<\/p>\n\n\n\n
sigmaE2 <-<\/span> sum<\/span>((<\/span>y-<\/span>y.hat)<\/span>^<\/span>2<\/span>)<\/span>\/<\/span>(<\/span>n-<\/span>p-<\/span>1<\/span>)<\/span>\nV.beta <-<\/span> sigmaE2*<\/span>solve<\/span>(<\/span>t<\/span>(<\/span>X)<\/span>%*%<\/span>X)<\/span>\n<\/code><\/pre>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a<\/h3>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u306e\u691c\u5b9a\u306f\u3001\u56de\u5e30\u4fc2\u6570\u306e\u63a8\u5b9a\u5024\u3068\u56de\u5e30\u4fc2\u6570\u306e\u6a19\u6e96\u8aa4\u5dee\u306e\u6bd4\u3092\u691c\u5b9a\u7d71\u8a08\u91cf\u3068\u3057\u3066\u5b9f\u65bd\u3059\u308b\u3002<\/p>\n\n\n\n
\u56de\u5e30\u4fc2\u6570\u306e\u6a19\u6e96\u8aa4\u5dee\u306f\u3001\u5148\u307b\u3069\u8a08\u7b97\u3057\u305f\u5206\u6563\u5171\u5206\u6563\u884c\u5217\u306e\u5bfe\u89d2\u6210\u5206 diag() \u306e\u5e73\u65b9\u6839 sqrt() \u3060\u3002<\/p>\n\n\n\n
SE.beta <-<\/span> sqrt<\/span>(<\/span>diag<\/span>(<\/span>V.beta))<\/span>\nt <-<\/span> beta\/<\/span>SE.beta\npt<\/span>(<\/span>abs<\/span>(<\/span>t),<\/span> n-<\/span>p-<\/span>1<\/span>,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>*<\/span>2<\/span>\n<\/code><\/pre>\n\n\n\n
\u6a19\u6e96\u8aa4\u5dee\uff08SE.beta\uff09\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
><\/span> matrix<\/span>(<\/span>SE.beta)<\/span>\n[,<\/span>1<\/span>]<\/span>\n[<\/span>1<\/span>,]<\/span> 1.1176259965<\/span>\n[<\/span>2<\/span>,]<\/span> 0.0078395225<\/span>\n[<\/span>3<\/span>,]<\/span> 0.0034756701<\/span>\n[<\/span>4<\/span>,]<\/span> 0.0151219066<\/span>\n[<\/span>5<\/span>,]<\/span> 0.0006877272<\/span>\n[<\/span>6<\/span>,]<\/span> 0.0050520870<\/span>\n[<\/span>7<\/span>,]<\/span> 0.0060082977<\/span>\n[<\/span>8<\/span>,]<\/span> 0.0371257460<\/span>\n<\/code><\/pre>\n\n\n\n
t\u5024\u306f\u3001\u56de\u5e30\u4fc2\u6570\u3068\u6a19\u6e96\u8aa4\u5dee\u306e\u6bd4\u3067\u8a08\u7b97\u3055\u308c\u308b\u3002<\/p>\n\n\n\n
><\/span> t\n[,<\/span>1<\/span>]<\/span>\nIntercept     6.8964873<\/span>\nCompPrice    11.9796718<\/span>\nIncome        3.7033759<\/span>\nAdvertising   8.6539134<\/span>\nPopulation   -<\/span>0.1801952<\/span>\nPrice       -<\/span>18.3137403<\/span>\nAge          -<\/span>7.4853715<\/span>\nEducation    -<\/span>1.0770004<\/span>\n<\/code><\/pre>\n\n\n\n
\u6700\u5f8c\u306b\u6709\u610f\u78ba\u7387\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
><\/span> pt<\/span>(<\/span>abs<\/span>(<\/span>t),<\/span> n-<\/span>p-<\/span>1<\/span>,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>*<\/span>2<\/span>\n[,<\/span>1<\/span>]<\/span>\nIntercept   2.145154e-11<\/span>\nCompPrice   2.153866e-28<\/span>\nIncome      2.432641e-04<\/span>\nAdvertising 1.302560e-16<\/span>\nPopulation  8.570924e-01<\/span>\nPrice       1.409811e-54<\/span>\nAge         4.751078e-13<\/span>\nEducation   2.821424e-01<\/span>\n<\/code><\/pre>\n\n\n\n
\u3053\u308c\u3089\u306f\u3059\u3079\u3066lm() \u3092\u4f7f\u3063\u305f\u7d50\u679c\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n\n\n\n
><\/span> summary<\/span>(<\/span>lm2)$<\/span>coefficients\n               Estimate   Std. Error     t value     Pr<\/span>(<\/span>>|<\/span>t|<\/span>)<\/span>\n(<\/span>Intercept)<\/span>  7.7076934384<\/span> 1.1176259965<\/span>   6.8964873<\/span> 2.145154e-11<\/span>\nCompPrice    0.0939149066<\/span> 0.0078395225<\/span>  11.9796718<\/span> 2.153866e-28<\/span>\nIncome       0.0128717129<\/span> 0.0034756701<\/span>   3.7033759<\/span> 2.432641e-04<\/span>\nAdvertising  0.1308636707<\/span> 0.0151219066<\/span>   8.6539134<\/span> 1.302560e-16<\/span>\nPopulation  -<\/span>0.0001239252<\/span> 0.0006877272<\/span>  -<\/span>0.1801952<\/span> 8.570924e-01<\/span>\nPrice       -<\/span>0.0925226099<\/span> 0.0050520870<\/span> -<\/span>18.3137403<\/span> 1.409811e-54<\/span>\nAge         -<\/span>0.0449743402<\/span> 0.0060082977<\/span>  -<\/span>7.4853715<\/span> 4.751078e-13<\/span>\nEducation   -<\/span>0.0399844437<\/span> 0.0371257460<\/span>  -<\/span>1.0770004<\/span> 2.821424e-01<\/span>\n<\/code><\/pre>\n\n\n\n
\u91cd\u76f8\u95a2\u4fc2\u6570\u3068\u5bc4\u4e0e\u7387<\/h3>\n\n\n\n
\u6700\u5f8c\u306b\u3001\u91cd\u76f8\u95a2\u4fc2\u6570\u3068\u5bc4\u4e0e\u7387\uff08\u91cd\u76f8\u95a2\u4fc2\u6570\u306e\u4e8c\u4e57\uff09\u306e\u8a08\u7b97\u3092\u793a\u3059\u3002<\/p>\n\n\n\n
\u91cd\u56de\u5e30\u5f0f\u306e\u5f53\u3066\u306f\u307e\u308a\u306e\u826f\u3055\u3092\u8868\u3059\u6307\u6a19\u306e\u4e00\u3064\u304c\u5bc4\u4e0e\u7387\u3060\u3002<\/p>\n\n\n\n
\u91cd\u76f8\u95a2\u4fc2\u6570\u306f\u5bc4\u4e0e\u7387\u306e\u5e73\u65b9\u6839\u3067\u3001\u5b9f\u6e2c\u5024 y \u3068\u4e88\u6e2c\u5024 y.hat \u306e\u76f8\u95a2\u4fc2\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
\u91cd\u76f8\u95a2\u4fc2\u6570\u306f0.735976\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
><\/span> cor.test<\/span>(<\/span>y,<\/span>y.hat)<\/span>\nPearson's product-moment correlation<\/span>\ndata:  y and y.hat<\/span>\nt = 21.688, df = 398, p-value < 2.2e-16<\/span>\nalternative hypothesis: true correlation is not equal to 0<\/span>\n95 percent confidence interval:<\/span>\n 0.6875397 0.7778921<\/span>\nsample estimates:<\/span>\n     cor <\/span>\n0.735976 <\/span>\n<\/code><\/pre>\n\n\n\n
\u5bc4\u4e0e\u7387\u306f\u30010.5416606\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
><\/span> R2 <-<\/span> cor<\/span>(<\/span>y,<\/span>y.hat)<\/span>^<\/span>2<\/span>\n><\/span> R2\n[,<\/span>1<\/span>]<\/span>\n[<\/span>1<\/span>,]<\/span> 0.5416606<\/span>\n<\/code><\/pre>\n\n\n\n
\u5bc4\u4e0e\u7387\u304c\u30bc\u30ed\u3067\u306a\u3044\u306e\u3092\u78ba\u8a8d\u3059\u308b\u691c\u5b9a\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
F<\/span> <-<\/span> (<\/span>R2\/<\/span>p)<\/span>\/<\/span>((<\/span>1<\/span>-<\/span>R2)<\/span>\/<\/span>(<\/span>n-<\/span>p-<\/span>1<\/span>))<\/span>\npF <-<\/span> pf<\/span>(<\/span>F<\/span>,<\/span> p,<\/span> n-<\/span>p-<\/span>1<\/span>,<\/span> lower.tail=<\/span>FALSE<\/span>)<\/span>\n<\/code><\/pre>\n\n\n\n
\u7d50\u679c\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3002<\/p>\n\n\n\n
><\/span> F<\/span>\n[,<\/span>1<\/span>]<\/span>\n[<\/span>1<\/span>,]<\/span> 66.18021<\/span>\n><\/span> pF\n[,<\/span>1<\/span>]<\/span>\n[<\/span>1<\/span>,]<\/span> 1.413772e-62<\/span>\n<\/code><\/pre>\n\n\n\n
\u5bc4\u4e0e\u7387\u53ca\u3073\u5bc4\u4e0e\u7387\u306e\u691c\u5b9a\u7d50\u679c\u306f\u3001lm() \u306e summary()\u3067\u8868\u793a\u3055\u308c\u308b\u6700\u4e0b\u6bb5\u306e\u8a18\u8ff0\u306b\u73fe\u308c\u308b\u3002<\/p>\n\n\n\n
Multiple R-<\/span>squared:<\/span>  0.5417<\/span>,<\/span>    Adjusted R-<\/span>squared:<\/span>  0.5335<\/span>\nF<\/span>-<\/span>statistic:<\/span> 66.18<\/span> on 7<\/span> and 392<\/span> DF,<\/span>  p-<\/span>value:<\/span> <<\/span> 2.2e-16<\/span>\n<\/code><\/pre>\n\n\n\n
\u307e\u3068\u3081<\/h2>\n\n\n\n
\u91cd\u56de\u5e30\u5206\u6790\u3092 step by step \u3067 R \u3092\u7528\u3044\u3066\u3001\u884c\u5217\u8a08\u7b97\u3067\u5b9f\u884c\u3057\u3066\u307f\u305f\u3002<\/p>\n\n\n\n
\u53c2\u8003\u306b\u306a\u308c\u3070\u3002<\/p>\n\n\n\n
\u53c2\u8003\u66f8\u7c4d<\/h2>\n\n\n\n
\u6559\u79d1\u66f8\u306f\u3001\u300c\u533b\u5b66\u3078\u306e\u7d71\u8a08\u5b66\u300d<\/p>\n\n\n\n
15.3 \u91cd\u56de\u5e30\u5206\u6790<\/p>\n\n\n\n
\"\u533b\u5b66\u3078\u306e\u7d71\u8a08\u5b66<\/a><\/figure>\n\n\n\n
<\/div>\n\n\n\n
\n
\u533b\u5b66\u3078\u306e\u7d71\u8a08\u5b66 (\u7d71\u8a08\u30e9\u30a4\u30d6\u30e9\u30ea\u30fc)<\/a><\/p>\n